The rescaled partial-sum process of independent, identically distributed random variables with finite variance converges in distribution to Brownian motion. Proved by Monroe Donsker, it is the functional generalization of the central limit theorem, sometimes called the invariance principle.
Facts
StatementThe theorem extends the central limit theorem to the whole empirical distribution function, showing that a suitably rescaled empirical process built from independent identically distributed samples converges in distribution to a Brownian bridge. 1 Proof YearThe cited source describes a general extension of the Doob-Kolmogorov heuristic approach as stated and proved in 1952, following an earlier 1951 paper by the same author establishing the invariance principle itself. Connections
Sources
1. Donsker's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentence
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions.
history section, attribution sentence
In 1952 Donsker stated and proved (not quite correctly) a general extension for the Doob-Kolmogorov heuristic approach.
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